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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the growth of the integral means of subharmonic functions of order less than one


Author: Faruk F. Abi-Khuzam
Journal: Trans. Amer. Math. Soc. 241 (1978), 239-252
MSC: Primary 30A64; Secondary 31A05
MathSciNet review: 0481002
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Abstract: Let u be a subharmonic function of order $ \lambda (0 < \lambda < 1)$, and let $ {m_s}(r,u)\, = \,{\left\{ {(1/2\pi )\int_{ - \pi }^\pi {{{\left\vert {u(r{e^{i\theta }})} \right\vert}^s}} d\theta } \right\}^{1/s}}$. We compare the growth of $ {m_s}(r,u)$ with that of the Riesz mass of u as measured by $ N\,(r,u)\, = (1/2\pi )\int_{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }$. A typical result of this paper states that the following inequality is sharp:

$\displaystyle \underset{x\to \infty }{\mathop{\lim \,\inf }}\,\,\frac{{{m}_{s}}... ...N\left( r,\,u \right)}\,\leqslant\,{{m}_{s}}\left( {{\psi }_{\lambda }} \right)$ ($ \ast$)

where $ \psi_\lambda (\theta )\, = \,(\pi \lambda /\sin \,\lambda )\cos \,\lambda \theta $. The case $ s\, = \,1$ is due to Edrei and Fuchs, the case $ s\, = \,2$ is due to Miles and Shea and the case $ s\, = \,\infty $ is due to Valiron.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0481002-9
PII: S 0002-9947(1978)0481002-9
Keywords: Riesz mass, order of a subharmonic function, proximate order
Article copyright: © Copyright 1978 American Mathematical Society